Ren Guo
In this paper we give two different proofs of Bobenko and Springborn's theorem of circle pattern: there exists a hyperbolic (or Euclidean) circle pattern with proscribed intersection angles and cone angles on a cellular decomposed surface up to isometry (or similarity).
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We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex projective structure. Motivated by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending a complex projective structure admitting a circle packing with given nerve (resp. a Delaunay circle pattern with given nerve and intersection angles) to the underlying complex structure is proper.
Thurston's Circle Pattern Theorem studies existence and rigidity of circle patterns of a given combinatorial type and the given non-obtuse exterior intersection angles. Using topological degree theory, variational principle, Teichmuller theory, and Sard's Theorem, this paper generalizes Circle Pattern Theorem to the case of obtuse exterior intersection angles.
This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time behaviors of Chow-Luo's combinatorial Ricci flows for these patterns. As consequences, several generalizations of circle pattern theorem are obtained. Moreover, our approach suggests a computational method to find the desired circle patterns.
This paper studies circle patterns from the viewpoint of configurations. By using the topological degree theory, we extend the Koebe-Andreev-Thurston Theorem to include circle patterns with obtuse exterior intersection angles. As a consequence, we obtain a generalized Andreev's Theorem which allows obtuse dihedral angles.
Generalized circle patterns on surfaces with cusps
本文在带尖点曲面上证明了Guo-Luo广义圆模式的存在性，推广了Bobenko-Springborn的双曲圆模式，并引入组合Ricci流与Calabi流，证明其解长时间存在且收敛。
This paper proves a deformation circle pattern theorem, which gives a complete description of those circle patterns with interstices in terms of the combinatorial type, the exterior intersections angles and the conformal structures of interstices. As results, the surface version of Rivin's theorem and the approximation property of packable surfaces are obtained.
A note on the pure cactus group of degree three and the configuration space of four points on the circle
本文明确描述了三次纯仙人掌群与圆上四点构型空间之间的关系。
Circle Pattern Theorem for Quasi-simplicial Triangulated Surfaces
本文将圆模式定理扩展到拟简单三角形化表面，证明在满足KAT不等式条件下，曲率映射的结构与简单情况相同，拓展了圆模式的应用范围。
Generalizing Andreev's Theorem via circle patterns
本文推导出允许钝重叠角的扩展圆模式定理，刻画了具钝二面角的紧凸双曲多面体子类，从而推广了 Andreev 定理，并建立了“弱解/正则性理论”的离散类比以完成证明。
We define a notion of semi-conjugacy between orientation-preserving actions of a group on the circle, which for fixed point free actions coincides with a classical definition of Ghys. We then show that two circle actions are semi-conjugate if and only if they have the same bounded Euler class. This settles some existing confusion present in the literature.
A ``hyperideal circle pattern'' in $S^2$ is a finite family of oriented circles, similar to the ``usual'' circle patterns but such that the closed disks bounded by the circles do not cover the whole sphere. Hyperideal circle patterns are directly related to hyperideal hyperbolic polyhedra, and also to circle packings. To each hyperideal circle pattern, one can associate an incidence graph and a set of intersection angles. We characterize the possible incidence graphs and intersection angles of hyperideal circle patterns in the sphere, the torus, and in higher genus surfaces. It is a consequenc…
The Koebe-Andreev-Thurston circle packing theorem, as well as its generalization to circle patterns due to Bobenko and Springborn, holds for Euclidean and hyperbolic metrics possibly with conical singularities, but fails for spherical metrics because of the non-uniqueness coming from Möbius transformations. In this paper, we show that a unique existence result for circle pattern with spherical conical metric holds if one prescribes the geodesic total curvature of each circle instead of the cone angles.
We consider ``hyperideal'' circle patterns, i.e. patterns of disks appearing in the definition of the Delaunay decomposition associated to a set of disjoint disks, possibly with cone singularities at the center of those disks. Hyperideal circle patterns are associated to hyperideal hyperbolic polyhedra. We describe the possible intersection angles and singular curvatures of those circle patterns, on Euclidean or hyperbolic surfaces with conical singularities. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra.
A note on Stein fillability of circle bundles over symplectic manifolds
本文证明了维数 $2n \geq 4$ 的闭积分辛流形上，当整数 $k>\int_Σω^{n}$ 时，$(Σ, kω)$ 上的Boothby-Wang丛不带有Stein可填充切触结构，从而否定地回答了Eliashberg的问题，并证明了某些孤立奇点的不可平滑性。
Combinatorial Calabi flows with ideal circle patterns
本文将理想圆型的组合Ricci流推广到双曲与欧几里得背景下的组合Calabi流，证明解长期存在并指数收敛到相应几何度量。
A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the circles, whether it is unique and how it may be constructed. Somewhat more generally, we allow cone-like singularities in the centers and intersection points of the circles. We prove existence and uniqueness theorems for the solution of the circle pattern problem using a variationa…
Perron's method and spherical ideal circle patterns with prescribed total geodesic curvatures
本文利用经典佩罗方法证明了在球面锥形度量下，当交角测度在(0,π/2]范围内且圆的测地曲率被指定时，圆图案的存在性和唯一性。同时展示了图斯顿算法在该设定下的收敛性。
Space of circle patterns on tori and its symplectic form
本文研究闭环面上圆模式空间的辛几何与拓扑结构，证明其通过嵌入获得的 Weil-Petersson 辛形式具有非退化性，并证明该空间同胚于闭环面的 Teichmüller 空间。
We investigate the combinatorial Ricci flow on a surface of nonpositive Euler characteristic when the necessary and sufficient condition for the convergence of the combinatorial Ricci flow is not valid. This observation addresses one of questions raised by B. Chow and F. Luo.
Pullback of symplectic forms to the space of circle patterns
本文研究了复射影结构曲面上圆图案形变空间的两个辛形式拉回，证明Goldman辛形式与Weil-Petersson辛形式的拉回重合，借此证明形变空间光滑性，并猜想其拉回非退化以定义辛结构。
The Convergence of Prescribed Combinatorial Ricci Flows for Total Geodesic Curvatures in Spherical Background Geometry
针对球面背景几何中预定全测地曲率的圆模式度量问题，本文构造并分析预定组合 Ricci 流，证明其收敛性，并给出求解退化情形的两种方法。
Combinatorial Calabi flows for ideal circle patterns in spherical background geometry
本文首次在球面背景几何中引入组合Calabi流，用于寻找具有指定总测地曲率的理想圆模式，并证明其解全局存在且收敛当且仅当目标圆模式存在，收敛时具有指数速度。
The main objective of this study is to understand how geometric hyper-ideal circle patterns can be constructed from given combinatorial angle data. We design a hybrid method consisting of a topological/deformation approach augmented with a variational principle. In this way, together with the question of characterization of hyper-ideal patterns in terms of angle data, we address their constructability via convex optimization. We presents a new proof of the main results from Jean-Marc Schlenker's work on hyper-ideal circle patterns by developing an approach that is potentially more suitable for…
We establish a correspondence between the dimer model on a bipartite graph and a circle pattern with the combinatorics of that graph, which holds for graphs that are either planar or embedded on the torus. The set of positive face weights on the graph gives a set of global coordinates on the space of circle patterns with embedded dual. Under this correspondence, which extends the previously known isoradial case, the urban renewal (local move for dimer models) is equivalent to the Miquel move (local move for circle patterns). As a consequence the Miquel dynamics on circle patterns is governed b…
Infinite circle patterns in the Weil-Petersson class
本文研究了Weil-Petersson类中由离散调和函数参数化的无限圆图案空间，揭示了其与双曲体积泛函Hessian诱导的Riemann度量的关系。
With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral surfaces with non-positive curvature. We show that in the case of such surfaces discrete uniformization via hyper-ideal circle patterns always exists and is unique. We also propose a numerical algorithm, utilizing convex optimization, that constructs the desired discrete uniformization.
In this note, we investigate an infinite one parameter family of circle packings, each with a set of three mutually tangent circles. We use these to generate an infinite set of circle packings with the Apollonian property. That is, every circle in the packing is a member of a cluster of four mutually tangent circles.
We prove existence and uniqueness results for patterns of circles with prescribed intersection angles in constant curvature surfaces. Our method is based on two new functionals--one for the Euclidean and one for the hyperbolic case. We show how Colin de Verdi`ere's, Br"agger's and Rivin's functionals can be derived from ours.
Projective rigidity of circle patterns and polyhedral surfaces in hyperbolic ends
本文证明了亏格 g≥2 闭曲面上带指定组合与交角的 Delaunay 圆模式复射影结构空间是 6g-6 维流形，且遗忘映射为 Lagrangian 浸入，推广了 Bonsante-Wolf 的圆填充结果，并借助双曲端理想多面体曲面将结论拓展到指定边长及其他多面体类型。
In [12], the existence of ideal circle patterns in Euclidean or hyperbolic background geometry under the combinatorial conditions was proved using flow approaches. It remains as an open problem for the spherical case. In this paper, we introduce a combinatorial geodesic curvature flow in spherical background geometry, which is analogous to the combinatorial Ricci flow of Chow and Luo in [4]. We characterize the sufficient and necessary condition for the convergence of the flow. That is, the prescribed geodesic curvature satisfies certain geometric and combinatorial condition if and only if for…
Circle action of the punctured mapping class group and cross homomorphism
本文通过映射类群对圆的作用及其旋转数，给出了无限循环群生成元的新几何解释，并与Furuta和Trapp的构造方法等价。
Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns
本文研究有限与无限理想圆模式的组合 p 阶 Calabi 流，给出有限情形收敛的充要条件，并证明无限情形下解的长时间存在性。
CMC-1 surfaces via osculating Möbius transformations between circle patterns
本文研究平面内相同组合圆模式间的莫比乌斯变换，导出双曲空间中对偶图的实现，刻画其在共享剪切坐标或交角时的一一对应关系，类比常平均曲率曲面的 Weierstrass 表示，并在三角格上建立收敛性。
Given a triangulation of a closed surface, we consider a cross ratio system that assigns a complex number to every edge satisfying certain polynomial equations per vertex. Every cross ratio system induces a complex projective structure together with a circle pattern on the closed surface. In particular, there is an associated conformal structure. We show that for any triangulated torus, the projection from the space of cross ratio systems with prescribed Delaunay angles to the Teichmüller space is a covering map with at most one branch point. Our approach is based on a notion of discrete holom…
The character of ideal circle patterns
本文提出了一种新的特征型标准，用于判断理想圆图案的存在性，提供了更易于验证的条件，并应用于曲率图像集的描述。
Unity of Jones polynomials in the unit circle and the plane
本文研究Jones多项式等于1的解集，证明双扭结族中单位根（除-1外）满足Jones多项式等于1，且该解集在单位圆上稠密，并构造了在复平面上稠密的链接家族。
We give an overview of various counting problems for Apollonian circle packings, which turn out to be related to problems in dynamics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared for the 13th Takagi lectures given at RIMS, Kyoto in the fall of 2013.
Discrete minimal surfaces: Old and New
本文探讨了保持结构的离散极小曲面构造，通过平行面偏移定义，关联消失平均曲率与变分特征，揭示离散极小曲面与圆图案变形空间及Teichmüller理论的联系。
We say that a collection Gamma of geodesics in the hyperbolic plane H^2 is a modular pattern if Gamma is invariant under the modular group PSL_2(Z), if there are only finitely many PSL_2(Z)-equivalence classes of geodesics in Gamma, and if each geodesic in Gamma is stabilized by an infinite order subgroup of PSL_2(Z). For instance, any finite union of closed geodesics on the modular orbifold H^2/PSL_2(Z) lifts to a modular pattern. Let S^1 be the ideal boundary of H^2. Given two points p,q in S^1 we write pq if p and q are the endpoints of a geodesic in Gamma. (In particular pp.) We show that …
Stationary measures on the circle from hyperbolic surfaces with cusps cannot be straightened by quasi-symmetries
本文证明：由带尖点的有限面积双曲曲面基本群上随机游走产生的、满足尖点游走指数衰减的圆周平稳测度，其关于勒贝格测度的奇异性在拟对称映射下保持稳定，从而解决了McMullen猜想。
Let $M$ be a smooth connected orientable compact surface. Denote by $F(M,S^1)$ the space of all Morse functions $f:M\to S^1$ having no critical points on the boundary of $M$ and such that for every boundary component $V$ of $M$ the restriction $f|_{V}:V\to S^1$ is either a constant map or a covering map. Endow $F(M,S^1)$ with the $C^{\infty}$-topology. In this note the connected components of $F(M,S^1)$ are classified. This result extends the results of S. V. Matveev, V. V. Sharko, and the author for the case of Morse functions being locally constant on the boundary of $M$.
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